Math 120A HW 4

1. Recall that $\mathbb{Z}_n$ denotes the group $(\mathbb{Z}_n,+_n)$.
   1. Does $\mathbb{Z}_3$ have a nontrivial proper subgroup?
   2. Does $\mathbb{Z}_4$ have a nontrivial proper subgroup?
   For each part, justify your answer by defining a subset and showing that it is a nontrivial proper subgroup, or else by proving that no nontrivial proper subgroup can exist.
2. In this problem we consider intersections and unions of subgroups.
   1. Let $(G,\ast)$ be a group and let $H$ and $K$ be subgroups of $G$. Prove that the intersection $H \cap K$ is a subgroup of $G$. Hint: instead of using the definition of "subgroup", it is easier to use the equivalent formulation in terms of closure properties (Theorem 5.14 in the book.)
   2. Find subgroups $H$ and $K$ of $\mathbb{Z}_6$ such that the union $H \cup K$ is not a subgroup of $\mathbb{Z}_6$. (When you claim that something is or is not a subgroup, make sure to say why.)
3. For this problem only, you do not need to justify your answers.
   1. List all the subgroups of $\mathbb{Z}_4$ and draw them in a subgroup diagram.
   2. List all the subgroups of $\mathbb{Z}_5$ and draw them in a subgroup diagram.
   3. List all the subgroups of $\mathbb{Z}_6$ and draw them in a subgroup diagram.
4. Let $G$ and $G'$ be groups, let $\phi$ be a homomorphism from $G$ to $G'$, and let $a \in G$. Because we haven't given names to the operations of $G$ and $G'$, we default to using multiplicative notation.
   1. Let $n \in \mathbb{Z}$ and prove that $\phi(a^n) = \phi(a)^n$. Hint: for the case $n \lt 0$ you can use the fact that $\phi(a^{-1}) = \phi(a)^{-1}$. We only proved this for isomorphisms of groups, but the same proof can be seen to work for homomorphisms of groups if we recall the fact that $\phi(e_G) = e_{G'}$.
   2. Use part (a) to prove that $\phi[\langle a \rangle] = \langle \phi(a) \rangle$. Recall that $\langle a \rangle$ denotes the cyclic subgroup $\{ a^n : n \in \mathbb{Z}\}$ and $\phi[X]$ denotes the pointwise image $\{\phi(x) : x \in X\}$.
   3. Rewrite your proof in part (b) for the specific case that the first group is $(\mathbb{R}^+,\cdot)$, the second group is $(\mathbb{R},+)$, and the homomorphism is the log function. Note that you should now use additive notation for elements of the second group.
5. To understand a group it is helpful to know the orders of its elements, as we will see in this problem.
   1. Compute the order of each element of $\mathbb{Z}_{10}$.
   2. Use this to show that no subgroup of $\mathbb{Z}_{10}$ can be isomorphic to $\mathbb{Z}_3$. (Note: this is not as simple as showing that $\mathbb{Z}_3$ itself is not a subgroup of $\mathbb{Z}_{10}$. For example, $\mathbb{Z}_{10}$ has a subgroup isomorphic to $\mathbb{Z}_2$, namely $\{0,5\}$, even though $\mathbb{Z}_2$ itself is not a subgroup of $\mathbb{Z}_{10}$.)
6. Let $G$ be the group of all matrices of the form $\begin{pmatrix}a & 0\\0 & b\end{pmatrix}$ where $a,b\gt 0$, under the operation of matrix multiplication. Prove that $G$ is not cyclic.